On the parity of generalized partition functions, III
Journal de Théorie des Nombres de Bordeaux, Volume 22 (2010) no. 1, pp. 51-78.

Improving on some results of J.-L. Nicolas [15], the elements of the set 𝒜=𝒜(1+z+z 3 +z 4 +z 5 ), for which the partition function p(𝒜,n) (i.e. the number of partitions of n with parts in 𝒜) is even for all n6 are determined. An asymptotic estimate to the counting function of this set is also given.

Dans cet article, nous complétons les résultats de J.-L. Nicolas [15], en déterminant tous les éléments de l’ensemble 𝒜=𝒜(1+z+z 3 +z 4 +z 5 ) pour lequel la fonction de partition p(𝒜,n) (c-à-d le nombre de partitions de n en parts dans 𝒜) est paire pour tout n6. Nous donnons aussi un équivalent asymptotique à la fonction de décompte de cet ensemble.

Received:
Published online:
DOI: 10.5802/jtnb.704
Classification: 11P81,  11N25,  11N37
Keywords: Partitions, periodic sequences, order of a polynomial, orbits, 2-adic numbers, counting function, Selberg-Delange formula.
Fethi Ben Saïd 1; Jean-Louis Nicolas 2; Ahlem Zekraoui 1

1 Université de Monastir Faculté des Sciences de Monastir Avenue de l’Environement 5000 Monastir, Tunisie
2 Université de Lyon 1 Institut Camile Jordan, UMR 5208 Batiment Doyen Jean Braconnier 21 Avenue Claude Bernard F-69622 Villeurbanne, France
@article{JTNB_2010__22_1_51_0,
     author = {Fethi Ben Sa{\"\i}d and Jean-Louis Nicolas and Ahlem Zekraoui},
     title = {On the parity of generalized partition {functions,~III}},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {51--78},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {1},
     year = {2010},
     doi = {10.5802/jtnb.704},
     zbl = {1236.11088},
     mrnumber = {2675873},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.704/}
}
TY  - JOUR
TI  - On the parity of generalized partition functions, III
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2010
DA  - 2010///
SP  - 51
EP  - 78
VL  - 22
IS  - 1
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.704/
UR  - https://zbmath.org/?q=an%3A1236.11088
UR  - https://www.ams.org/mathscinet-getitem?mr=2675873
UR  - https://doi.org/10.5802/jtnb.704
DO  - 10.5802/jtnb.704
LA  - en
ID  - JTNB_2010__22_1_51_0
ER  - 
%0 Journal Article
%T On the parity of generalized partition functions, III
%J Journal de Théorie des Nombres de Bordeaux
%D 2010
%P 51-78
%V 22
%N 1
%I Université Bordeaux 1
%U https://doi.org/10.5802/jtnb.704
%R 10.5802/jtnb.704
%G en
%F JTNB_2010__22_1_51_0
Fethi Ben Saïd; Jean-Louis Nicolas; Ahlem Zekraoui. On the parity of generalized partition functions, III. Journal de Théorie des Nombres de Bordeaux, Volume 22 (2010) no. 1, pp. 51-78. doi : 10.5802/jtnb.704. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.704/

[1] N. Baccar, Sets with even partition functions and 2-adic integers. Periodica Math. Hung. 55 (2) (2007), 177–193. | MR: 2375042 | Zbl: 1164.11066

[2] N. Baccar and F. Ben Saïd, On sets such that the partition function is even from a certain point on. International Journal of Number Theory 5 n o 3 (2009), 407–428. | MR: 2529082 | Zbl: 05589358

[3] N. Baccar, F. Ben Saïd and A. Zekraoui, On the divisor function of sets with even partition functions. Acta Math. Hungarica 112 (1-2) (2006), 25–37. | MR: 2251128 | Zbl: 1121.11071

[4] F. Ben Saïd, On a conjecture of Nicolas-Sárközy about partitions. Journal of Number Theory 95 (2002), 209–226. | MR: 1924098 | Zbl: 1041.11066

[5] F. Ben Saïd, On some sets with even valued partition function. The Ramanujan Journal 9 (2005), 63–75. | MR: 2166378 | Zbl: 1145.11073

[6] F. Ben Saïd and J.-L. Nicolas, Sets of parts such that the partition function is even. Acta Arithmetica 106 (2003), 183–196. | EuDML: 278790 | MR: 1958984 | Zbl: 1052.11069

[7] F. Ben Saïd and J.-L. Nicolas, Sur une application de la formule de Selberg-Delange. Colloquium Mathematicum 98 n o 2 (2003), 223–247. | EuDML: 284796 | MR: 2033110 | Zbl: 1051.11049

[8] F. Ben Saïd and J.-L. Nicolas, Even partition functions. Séminaire Lotharingien de Combinatoire 46 (2002), B 46i (http//www.mat.univie.ac.at/ slc/). | EuDML: 123676 | MR: 1921679 | Zbl: 1042.11008

[9] F. Ben Saïd, H. Lahouar and J.-L. Nicolas, On the counting function of the sets of parts such that the partition function takes even values for n large enough. Discrete Mathematics 306 (2006), 1089–1096. | MR: 2245637 | Zbl: 1109.05019

[10] P. M. Cohn, Algebra, Volume 1, Second Edition. John Wiley and Sons Ltd, 1988). | MR: 663370

[11] H. Halberstam and H.-E. Richert, Sieve methods. Academic Press, New York, 1974. | MR: 424730 | Zbl: 0298.10026

[12] H. Lahouar, Fonctions de partitions à parité périodique. European Journal of Combinatorics 24 (2003), 1089–1096. | MR: 2024560 | Zbl: 1049.11110

[13] R. Lidl and H. Niederreiter, Introduction to finite fields and their applications. Cambridge University Press, revised edition, 1994. | MR: 1294139 | Zbl: 0820.11072

[14] J.-L. Nicolas, I.Z. Ruzsa and A. Sárközy, On the parity of additive representation functions. J. Number Theory 73 (1998), 292–317. | MR: 1657968 | Zbl: 0921.11050

[15] J.-L. Nicolas, On the parity of generalized partition functions II. Periodica Mathematica Hungarica 43 (2001), 177–189. | MR: 1830575 | Zbl: 0980.11049

Cited by Sources: